"Heat kernel analysis and generalized Ricci lower bounds in sub-Riemannian geometry" Fabrice Baudoin (Purdue Univ.)
Abstract
We study a class of rank two sub-Riemannian manifolds
encompassing Riemannian manifolds, CR manifolds with vanishing
Webster-Tanaka torsion, orthonormal bundles over Riemannian manifolds,
and graded nilpotent Lie groups of step two. These manifolds admit a
canonical horizontal connection and a canonical sub-Laplacian. We
construct on these manifolds an analogue of the Riemannian Ricci tensor
and prove Bochner type formulas for the sub-Laplacian. As a consequence,
it is possible to formulate on these spaces a sub-Riemannian analogue of
the so-called curvature dimension inequality. The heat kernel analysis
on sub-Riemannian manifolds for which this inequality is satisfied is
shown to share many properties in common with the heat kernel analysis
on Riemannian manifolds whose Ricci curvature is bounded from below.
This is mainly a joint work with N. Garofalo.